Garischismic clan: Difference between revisions

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This clan of temperaments tempers out the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 33554432/33480783, and includes these:
{{Technical data page}}
* [[Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783), the amount by which the [[Pythagorean comma]] falls short of the [[septimal comma]], thus equating the two.
* ''[[Newt]]'', {2401/2400, 33554432/33480783} → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]
* ''[[Quintagar]]'', {3136/3125, 33554432/33480783} → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Vulture]]'', {4375/4374, 33554432/33480783} → [[Vulture family #Vulture|Vulture family]]
* ''[[Trident]]'', {6144/6125, 14348907/14336000} → [[Tricot family #Trident|Tricot family]]
* [[Cotoneum]], {10976/10935, 823543/819200} → [[Hemimage temperaments #Cotoneum|Hemimage temperaments]]
* ''[[Paramity]]'', {65625/65536, 1600000/1594323} → [[Amity family #Paramity|Amity family]]
* ''[[Garistearn]]'', {118098/117649, 33554432/33480783} → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]
* ''[[Sextile]]'', {250047/250000, 33554432/33480783} → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]]
* ''[[Satin]]'', {2100875/2097152, 4802000/4782969} → [[Canousmic temperaments #Satin|Canousmic temperaments]]


== Gary ==
== Gary ==
Subgroup: 2.3.7
Gary, the head of this clan, may be viewed as the [[2.3.7 subgroup|2.3.7-subgroup]] counterpart of [[schismic]]. It is generated by a [[3/2|perfect fifth]], and 7/4 is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma. Gary can therefore use [[chain-of-fifths notation]] with an additional set of accidentals such as arrows to represent the comma step.
 
Just as there is the 1/8-schisma tuning for schismic, there is the 1/14-schisma tuning for gary, which tunes 7/4 pure by sharpening the perfect fifth by about 0.272 cents. Similarly, the 1/15-schisma tuning tunes [[7/6]] pure, and the 2/29-schisma tuning splits their difference, tuning the septimal diesis of [[49/48]] pure. [[135edo]] is close to the 1/14-schisma tuning, whereas [[634edo]] gives a tuning practically identical to 1/15-schisma. Other notable tunings not appearing in the optimal ET sequence include [[311edo]] and [[323edo]].
 
[[Subgroup]]: 2.3.7


[[Comma list]]: 33554432/33480783
[[Comma list]]: 33554432/33480783


[[Sval]] [[mapping]]: [{{val| 1 2 -3 }}, {{val| 0 -1 14 }}]
{{Mapping|legend=2| 1 0 25 | 0 1 -14 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9155{{c}}, ~3/2 = 702.1584{{c}}
: [[error map]]: {{val| -0.085 +0.119 +0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}
: error map: {{val| 0.000 +0.257 +0.201 }}
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd }}
 
[[Badness]] (Sintel): 0.463
 
=== Overview to extensions ===
==== Full 11-limit extensions ====
The second comma of the comma list determines which full 7-limit or 11-limit family member we are looking at. Garibaldi adds the [[schisma]], or equivalently [[225/224]] and finds 5/4 at the diminished fourth. Cotoneum adds [[10976/10935]] and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary.
 
Newt adds [[2401/2400]], halving the fifth. Gariwizmic adds the [[wizma]] with a 1/2-octave period. Alphatrident adds [[6144/6125]], slicing the twelfth in three. Satin adds [[2100875/2097152]], slicing the fourth in three. Vulture adds [[4375/4374]], slicing the twelfth in four. Sextile adds [[250047/250000]] with a 1/6-octave period. World calendar adds [[390625/388962]] with a 1/4-octave period as well as a halved fifth. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five. Heptacot adds [[703125/702464]], slicing the fifth in seven. Finally, garitritonic adds 95703125/95551488 ({{monzo| -17 -6 9 2 }}), slicing the 24th harmonic in nine.


[[POTE generator]]: ~3/2 = 702.2079
Temperaments discussed elsewhere are:
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* ''[[Alphatrident]]'' (+6144/6125) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Vulture]]'' (+4375/4374) → [[Vulture family #Vulture|Vulture family]]
* ''[[Quintagar]]'' (+3136/3125) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Paramity]]'' (+65625/65536) → [[Amity family #Paramity|Amity family]]
* ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]


{{Val list|legend=1| 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd }}
Considered below are cotoneum, newt, gariwizmic, satin, sextile, and world calendar.


[[Badness]]: 0.0135
==== Subgroup extensions ====
Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)<sup>2</sup> with 33/32, at the cost of doubling the complexity.  


=== 2.3.7.11 ===
=== 2.3.7.11 subgroup ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Comma list: 19712/19683, 41503/41472
Comma list: 19712/19683, 41503/41472


Sval mapping: [{{val| 1 2 -3 13 }}, {{val| 0 -1 14 -23 }}]
Subgroup-val mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
 
Optimal tunings:
* WE: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
 
Badness (Sintel): 0.276
 
== Cotoneum ==
{{Main| Cotoneum }}
: ''For the 5-limit version, see [[Schismic–countercommatic equivalence continuum #Cotoneum (5-limit)]].''
 
Cotoneum tempers out 10976/10935 ([[hemimage comma]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. This temperament can be described as {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found -49 generators away. In terms of chain-of-fifths notation, this is a sextuple-diminished octave, or a perfect fourth minus four generic commas. 
 
However, cotoneum can be notated like [[cassaschismic]], where 5/4 is conceptualized as an aberschisma-up comma-down major third (C–^↓E), but with the extra equivalence that the generic aberschisma is identical to the [[41-comma]]. In other words, we have C–^↑↑E = C–↓↓E.
 
It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 823543/819200
 
{{Mapping|legend=1| 1 0 80 25 | 0 1 -49 -14 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0386{{c}}, ~3/2 = 702.3396{{c}}
: [[error map]]: {{val| +0.039 +0.423 +0.244 -1.155 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3164{{c}}
: error map: {{val| 0.000 +0.361 +0.182 -1.256 }}
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~4/3 = [497.14286, 498.46154] (29\70 to 27\65)
* 9-odd-limit diamond monotone: ~4/3 = [497.14286, 498.11321] (29\70 to 22\53)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~4/3 = [497.64251, 498.04500]
 
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 258, 475 }}
 
[[Badness]] (Sintel): 2.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 10976/10935, 16384/16335
 
Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}
 
Optimal tunings:
* WE: ~2 = 1199.8629{{c}}, ~3/2 = 702.2224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3036{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~4/3 = [497.56098, 497.87234] (17\41 to 39\94)
* 11-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]
 
{{Optimal ET sequence|legend=0| 41, 135c, 176, 217 }}
 
Badness (Sintel): 1.68
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3584/3575, 10976/10935
 
Mapping: {{mapping| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }}
 
Optimal tunings:
* WE: ~2 = 1199.8897{{c}}, ~3/2 = 702.2415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3061{{c}}
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~4/3 = [497.56098, 497.77778] (17\41 to 56\135)
* 13-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]
* 15-odd-limit diamond tradeoff: ~4/3 = [497.63067, 498.04500]
 
{{Optimal ET sequence|legend=0| 41, 176, 217 }}
 
Badness (Sintel): 1.53
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
 
Mapping: {{mapping| 1 0 80 25 -33 -93 -137 | 0 1 -49 -14 23 61 89 }}
 
Optimal tunings:
* WE: ~2 = 1199.8939{{c}}, ~3/2 = 702.2445{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3064{{c}}
 
Tuning ranges:
* 17-odd-limit diamond monotone: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
* 17-odd-limit diamond tradeoff: ~4/3 = [497.63067, 498.04500]
 
{{Optimal ET sequence|legend=0| 41, 176, 217 }}
 
Badness (Sintel): 1.50
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
 
Mapping: {{mapping| 1 0 80 25 -33 -93 -137 74 | 0 1 -49 -14 23 61 89 -44 }}
 
Optimal tunings:
* WE: ~2 = 1199.8766{{c}}, ~3/2 = 702.2355{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3077{{c}}
 
Tuning ranges:
* 19- and 21-odd-limit diamond monotone: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
* 19- and 21-odd-limit diamond tradeoff: ~4/3 = [497.62290, 498.04500]
 
{{Optimal ET sequence|legend=0| 41, 176, 217 }}
 
Badness (Sintel): 1.33
 
== Newt ==
: ''For the 5-limit version, see [[Schismic–countercommatic equivalence continuum #Newt (5-limit)]].''
 
Newt tempers out the [[breedsma]] and may be described as the {{nowrap| 41 & 270 }} temperament. It has a generator of a neutral third (0.2 cents flat of [[49/40]]) with a [[ploidacot]] signature of dicot. 41 generator steps fall short of 12 octaves by a generic aberschisma step of a [[schisma]]~[[aberschisma]]. From there the intervals of 5 and 7 can be derived.
 
Like [[#Cotoneum|cotoneum]], newt can be notated in the same way as [[cassaschismic]], but with half-sharps and half-flats and the extra equivalence that two comma steps and an aberschisma step make a half-apotome step. In other words, C–^↑↑E = C–v↓↓E = C–Ed.
 
Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the lehmerisma ([[3025/3024]]). This extends into a very strong [[13-limit]] temperament and eventually a very strong no-17 [[19-limit]] temperament, a.k.a. ''neonewt''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] are more accurate.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 33554432/33480783
 
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9315{{c}}, ~49/40 = 351.0932{{c}}
: [[error map]]: {{val| -0.068 +0.163 +0.075 -0.188 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1141{{c}}
: error map: {{val| 0.000 +0.273 +0.180 -0.022 }}
 
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
 
[[Badness]] (Sintel): 1.06
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 19712/19683
 
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
 
Optimal tunings:
* WE: ~2 = 1199.9603{{c}}, ~49/40 = 351.1038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1155{{c}}
 
{{Optimal ET sequence|legend=0| 41, 188, 229, 270, 581, 851, 1121, 1972 }}
 
Badness (Sintel): 0.643
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9747{{c}}, ~49/40 = 351.1094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1168{{c}}
 
{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851, 2283b }}
 
Badness (Sintel): 0.571
 
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
 
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
 
Optimal tunings:
* WE: ~2 = 1199.9782{{c}}, ~49/40 = 351.1102{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1166{{c}}
 
{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851 }}
 
Badness (Sintel): 0.438
 
== Gariwizmic ==
Gariwizmic tempers out the [[wizma]] and the garischisma, and may be described as the {{nowrap| 94 & 176 }} temperament. It assumes a [[semioctave]] period and a [[3/2|perfect fifth]] generator that is slightly sharp of just. It finds [[5/4]] 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup. Notable tunings not appearing in the optimal ET sequence include [[364edo]] and [[634edo]].
 
Gariwizmic was named by [[Eufalesio]] in 2026 as a concatenation of ''gary'' and ''wizmic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 420175/419904, 33554432/33480783
 
{{Mapping|legend=1| 2 0 -119 50 | 0 1 39 -14 }}
: mapping generators: ~46305/32768, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~46305/32768 = 599.9657{{c}}, ~3/2 = 702.1765{{c}}
: [[error map]]: {{val|-0.069 +0.153 -0.021 -0.053 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~3/2 = 702.2161{{c}}
: error map: {{val| 0.000 +0.261 +0.114 +0.149 }}
 
{{Optimal ET sequence|legend=1| 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd }}
 
[[Badness]] (Sintel): 2.22
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 19712/19683, 41503/41472
 
Mapping: {{mapping| 2 0 -119 50 -66 | 0 1 39 -14 23 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9790{{c}}, ~3/2 = 702.1938{{c}}
: error map: {{val| -0.042 +0.197 +0.106 -0.001 -0.440 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2179{{c}}
: error map: {{val| 0.000 +0.263 +0.185 +0.123 -0.306 }}
 
{{Optimal ET sequence|legend=0| 94, 176, 270, 1174, 1444, 1714, 1984e }}
 
Badness (Sintel): 1.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683
 
Mapping: {{mapping| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9958{{c}}, ~3/2 = 702.2096{{c}}
: error map: {{val| -0.008 +0.246 +0.035 +0.146 -0.412 -0.353 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2145{{c}}
: error map: {{val| 0.000 +0.260 +0.054 +0.170 -0.383 -0.321 }}
 
{{Optimal ET sequence|legend=0| 94, 176, 270, 634, 904, 1174 }}
 
Badness (Sintel): 0.822
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079
 
Mapping: {{mapping| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9969{{c}}, ~3/2 = 702.2114{{c}}
: error map: {{val| -0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2150{{c}}
: error map: {{val| 0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055 }}
 
{{Optimal ET sequence|legend=0| 94, 176, 270, 634, 904, 1174 }}
 
Badness (Sintel): 0.655
 
== Satin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Satin]].''
 
Satin tempers out the [[rainy comma]] and the [[canousma]] in addition to the garischisma, and may be described as the {{nowrap| 94 & 217 }} temperament. It uses [[~]][[11/10]] as a generator, three of which gives a [[4/3|perfect fourth]], tempering out [[4000/3993]] in the 11-limit and onwards. Its [[ploidacot]] is omega-tricot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2100875/2097152, 4802000/4782969
 
{{Mapping|legend=1| 1 2 12 -3 | 0 -3 -70 42 }}
: mapping generators: ~2, ~8575/7776
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0198{{c}}, ~8575/7776 = 165.9161{{c}}
: [[error map]]: {{val| +0.020 +0.336 -0.200 -0.411 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8575/7776 = 165.9133{{c}}
: error map: {{val| 0.000 +0.305 -0.241 -0.469 }}
 
{{Optimal ET sequence|legend=1| 94, 217, 311, 839, 1150 }}
 
[[Badness]] (Sintel): 4.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 19712/19683, 41503/41472
 
Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}
 
Optimal tunings:
* WE: ~2 = 1199.9931{{c}}, ~11/10 = 165.9145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9155{{c}}
 
{{Optimal ET sequence|legend=0| 94, 217, 311 }}
 
Badness (Sintel): 1.92
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
 
Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}
 
Optimal tunings:
* WE: ~2 = 1199.9607{{c}}, ~11/10 = 165.9085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9141{{c}}
 
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}
 
Badness (Sintel): 1.25
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
 
Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}
 
Optimal tunings:
* WE: ~2 = 1199.9843{{c}}, ~11/10 = 165.9110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9132{{c}}
 
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}
 
Badness (Sintel): 1.02
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
 
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}
 
Optimal tunings:
* WE: ~2 = 1199.9875{{c}}, ~11/10 = 165.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9129{{c}}
 
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}
 
Badness (Sintel): 0.881
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
 
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}
 
Optimal tunings:
* WE: ~2 = 1199.9745{{c}}, ~11/10 = 165.9103{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9140{{c}}
 
{{Optimal ET sequence|legend=0| 94, 217, 311 }}
 
Badness (Sintel): 0.871
 
== Sextile ==
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Sextile (5-limit)]].''
 
Sextile tempers out the [[landscape comma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, 33554432/33480783
 
{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
: mapping generators: ~4096/3645, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}
 
{{Optimal ET sequence|legend=1| 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc }}
 
[[Badness]] (Sintel): 1.77
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 5632/5625, 9801/9800, 151263/151250
 
Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 258e, 270, 822, 1092, 1362c }}
 
Badness (Sintel): 0.981
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
 
Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 258ef, 270, 552, 822, 1092, 1914cde }}
 
Badness (Sintel): 0.788
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625
 
Mapping: {{mapping| 6 0 71 150 230 279 35 | 0 1 -6 -14 -22 -27 -1 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9826{{c}}, ~3/2 = 702.1359{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2003{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 258ef, 270, 552, 822, 1092 }}
 
Badness (Sintel): 0.634
 
== World calendar ==
World calendar tempers out the [[dimcomp comma]] and the garischisma, and can be described as the {{nowrap| 12 & 364 }} temperament. The name derives from a {{w|World Calendar|certain calendar layout}} by the same name.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 390625/388962, 33554432/33480783
 
{{Mapping|legend=1| 4 1 -44 86 | 0 2 -13 -28 }}
: mapping generators: ~25/21, ~91125/57344
 
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.9938{{c}}, ~91125/57344 = 801.0780{{c}}
: [[error map]]: {{val| -0.025 +0.195 -0.603 +0.452 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~91125/57344 = 801.0955{{c}}
: error map: {{val| 0.000 +0.236 -0.555 +0.501 }}
 
{{Optimal ET sequence|legend=1| 12, …, 352, 364 }}
 
[[Badness]] (Sintel): 7.39
 
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17
 
Comma list: 2025/2023, 24576/24565, 390625/388962
 
Subgroup-val mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}
 
Optimal tunings:
* WE: ~25/21 = 299.9861{{c}}, ~27/17 = 801.0536{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/17 = 801.0919{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 352, 364 }}
 
Badness (Sintel): 2.74
 
=== 2.3.5.7.17.19 subgroup ===
Subgroup: 2.3.5.7.17.19
 
Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000
 
Subgroup-val mapping: {{mapping| 4 1 -44 86 3 25 | 0 2 -13 -28 5 -3 }}
 
Optimal tunings:
* WE: ~25/21 = 299.9982{{c}}, ~27/17 = 801.0898{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/17 = 801.0946{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 352, 364 }}
 
Badness (Sintel): 1.82
 
== Heptacot ==
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Heptacot (5-limit)]].''
 
Heptacot tempers out the [[meter]] and may be described as the {{nowrap| 12 & 311 }} temperament, splitting the perfect fifth into seven semitones. It is the natural 7-limit extension of the 5-limit temperament named by [[Tristan Bay]] in 2024. [[311edo]] and [[323edo]] are obvious tuning choices, as well as anything in between such as [[634edo]].
 
Heptacot extends to the 11-limit in the same way as does gary, which best preserves its accuracy, though it should be noted that {{nowrap| 299 & 311 }} and {{nowrap| 323 & 335d }} make for simpler but less accurate alternative extensions.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 703125/702464, 33554432/33480783
 
{{Mapping|legend=1| 1 1 6 11 | 0 7 -44 -98 }}
: mapping generators: ~2, ~1323/1250
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9434{{c}}, ~1323/1250 = 100.3096{{c}}
: [[error map]]: {{val| -0.057 +0.155 -0.274 +0.215 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1250 = 100.3148{{c}}
: error map: {{val| 0.000 +0.249 -0.165 +0.324 }}
 
{{Optimal ET sequence|legend=1| 12, …, 299, 311, 323, 634, 957, 1591 }}
 
[[Badness]] (Sintel): 3.06
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 19712/19683, 41503/41472, 703125/702464
 
Mapping: {{mapping| 1 1 6 11 -10 | 0 7 -44 -98 161 }}
: mapping generators: ~2, ~1323/1250
 
Optimal tunings:
* WE: ~2 = 1199.9981{{c}}, ~1323/1250 = 100.3174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1323/1250 = 100.3176{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 311, 634, 945 }}
 
Badness (Sintel): 3.21
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 4096/4095, 19712/19683, 31250/31213
 
Mapping: {{mapping| 1 1 6 11 -10 -7 | 0 7 -44 -98 161 128 }}
: mapping generators: ~2, ~1323/1250
 
Optimal tunings:
* WE: ~2 = 1199.9938{{c}}, ~675/637 = 100.3169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~675/637 = 100.3174{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 311, 634, 945 }}
 
Badness (Sintel): 1.89
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 31250/31213
 
Mapping: {{mapping| 1 1 6 11 -10 -7 5 | 0 7 -44 -98 161 128 -9 }}
: mapping generators: ~2, ~1323/1250
 
Optimal tunings:
* WE: ~2 = 1200.0076{{c}}, ~675/637 = 100.3179{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~675/637 = 100.3173{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 311, 634, 945 }}
 
Badness (Sintel): 1.38
 
== Garitritonic ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
 
Garitritonic may be described as the {{nowrap| 53 & 581 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[634edo]] makes for a strong 7-limit tuning, though in the higher limits one may prefer sticking to [[581edo]].
 
Garitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''gary'' and ''tritonic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 33554432/33480783, 95703125/95551488
 
{{Mapping|legend=1| 1 -3 -15 67 | 0 9 34 -126 }}
: mapping generators: ~2, ~4375/3072
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9678{{c}}, ~4375/3072 = 611.3417{{c}}
: [[error map]]: {{val| -0.032 +0.217 -0.213 -0.036 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4375/3072 = 611.3582{{c}}
: error map: {{val| 0.000 +0.268 -0.136 +0.045 }}
 
{{Optimal ET sequence|legend=1| 53, 422d, 475, 528, 581, 634, 1215 }}
 
[[Badness]] (Sintel): 6.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 19712/19683, 41503/41472, 1953125/1948617
 
Mapping: {{mapping| 1 -3 -15 67 -102 | 0 9 34 -126 207 }}
 
Optimal tunings:
* WE: ~2 = 1199.9795{{c}}, ~4375/3072 = 611.3485{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~4375/3072 = 611.3589{{c}}
 
{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}
 
Badness (Sintel): 3.60
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 4096/4095, 19712/19683, 78125/78078
 
Mapping: {{mapping| 1 -3 -15 67 -102 -34 | 0 9 34 -126 207 74 }}
 
Optimal tunings:
* WE: ~2 = 1199.9813{{c}}, ~500/351 = 611.3494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3589{{c}}
 
{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}
 
Badness (Sintel): 1.73
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 59375/59319
 
Mapping: {{mapping| 1 -3 -15 67 -102 -34 -36 | 0 9 34 -126 207 74 79 }}


POTE generator: ~3/2 = 702.2292
Optimal tunings:
* WE: ~2 = 1199.9884{{c}}, ~500/351 = 611.3531{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3590{{c}}


Vals: {{Val list| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}


Badness: 0.00731
Badness (Sintel): 1.22


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:37, 20 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The garischismic clan of temperaments tempers out the garischisma (monzo: [25 -14 0 -1⟩, ratio: 33554432/33480783), the amount by which the Pythagorean comma falls short of the septimal comma, thus equating the two.

Gary

Gary, the head of this clan, may be viewed as the 2.3.7-subgroup counterpart of schismic. It is generated by a perfect fifth, and 7/4 is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma. Gary can therefore use chain-of-fifths notation with an additional set of accidentals such as arrows to represent the comma step.

Just as there is the 1/8-schisma tuning for schismic, there is the 1/14-schisma tuning for gary, which tunes 7/4 pure by sharpening the perfect fifth by about 0.272 cents. Similarly, the 1/15-schisma tuning tunes 7/6 pure, and the 2/29-schisma tuning splits their difference, tuning the septimal diesis of 49/48 pure. 135edo is close to the 1/14-schisma tuning, whereas 634edo gives a tuning practically identical to 1/15-schisma. Other notable tunings not appearing in the optimal ET sequence include 311edo and 323edo.

Subgroup: 2.3.7

Comma list: 33554432/33480783

Subgroup-val mapping: [⟨1 0 25], ⟨0 1 -14]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.9155 ¢, ~3/2 = 702.1584 ¢
error map: ⟨-0.085 +0.119 +0.027]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2124 ¢
error map: ⟨0.000 +0.257 +0.201]

Optimal ET sequence: 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd

Badness (Sintel): 0.463

Overview to extensions

Full 11-limit extensions

The second comma of the comma list determines which full 7-limit or 11-limit family member we are looking at. Garibaldi adds the schisma, or equivalently 225/224 and finds 5/4 at the diminished fourth. Cotoneum adds 10976/10935 and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary.

Newt adds 2401/2400, halving the fifth. Gariwizmic adds the wizma with a 1/2-octave period. Alphatrident adds 6144/6125, slicing the twelfth in three. Satin adds 2100875/2097152, slicing the fourth in three. Vulture adds 4375/4374, slicing the twelfth in four. Sextile adds 250047/250000 with a 1/6-octave period. World calendar adds 390625/388962 with a 1/4-octave period as well as a halved fifth. Quintagar adds 3136/3125, slicing the fourth in five. Paramity adds 65625/65536, slicing the eleventh in five. Heptacot adds 703125/702464, slicing the fifth in seven. Finally, garitritonic adds 95703125/95551488 ([-17 -6 9 2⟩), slicing the 24th harmonic in nine.

Temperaments discussed elsewhere are:

Considered below are cotoneum, newt, gariwizmic, satin, sextile, and world calendar.

Subgroup extensions

Gary can be naturally extended into the no-5's 11-limit with good accuracy by equating (64/63)2 with 33/32, at the cost of doubling the complexity.

2.3.7.11 subgroup

Subgroup: 2.3.7.11

Comma list: 19712/19683, 41503/41472

Subgroup-val mapping: [⟨1 0 25 -33], ⟨0 1 -14 23]]

Optimal tunings:

  • WE: ~2 = 1199.9631 ¢, ~3/2 = 702.2077 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2290 ¢

Optimal ET sequence: 12e, 41, 94, 135, 716, 851, 986, 1121, 1256

Badness (Sintel): 0.276

Cotoneum

Main article: Cotoneum
For the 5-limit version, see Schismic–countercommatic equivalence continuum #Cotoneum (5-limit).

Cotoneum tempers out 10976/10935 (hemimage comma), and 823543/819200 (quince comma) in addition to the garischisma. This temperament can be described as 41 & 217, and is supported by 176-, 217-, and 258edo. 5/4 is found -49 generators away. In terms of chain-of-fifths notation, this is a sextuple-diminished octave, or a perfect fourth minus four generic commas.

However, cotoneum can be notated like cassaschismic, where 5/4 is conceptualized as an aberschisma-up comma-down major third (C–^↓E), but with the extra equivalence that the generic aberschisma is identical to the 41-comma. In other words, we have C–^↑↑E = C–↓↓E.

It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 823543/819200

Mapping: [⟨1 0 80 25], ⟨0 1 -49 -14]]

Optimal tunings:

  • WE: ~2 = 1200.0386 ¢, ~3/2 = 702.3396 ¢
error map: ⟨+0.039 +0.423 +0.244 -1.155]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3164 ¢
error map: ⟨0.000 +0.361 +0.182 -1.256]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~4/3 = [497.14286, 498.46154] (29\70 to 27\65)
  • 9-odd-limit diamond monotone: ~4/3 = [497.14286, 498.11321] (29\70 to 22\53)
  • 7- and 9-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]

Optimal ET sequence: 41, 135c, 176, 217, 258, 475

Badness (Sintel): 2.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935, 16384/16335

Mapping: [⟨1 0 80 25 -33], ⟨0 1 -49 -14 23]]

Optimal tunings:

  • WE: ~2 = 1199.8629 ¢, ~3/2 = 702.2224 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3036 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~4/3 = [497.56098, 497.87234] (17\41 to 39\94)
  • 11-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]

Optimal ET sequence: 41, 135c, 176, 217

Badness (Sintel): 1.68

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3584/3575, 10976/10935

Mapping: [⟨1 0 80 25 -33 -93], ⟨0 1 -49 -14 23 61]]

Optimal tunings:

  • WE: ~2 = 1199.8897 ¢, ~3/2 = 702.2415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3061 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~4/3 = [497.56098, 497.77778] (17\41 to 56\135)
  • 13-odd-limit diamond tradeoff: ~4/3 = [497.64251, 498.04500]
  • 15-odd-limit diamond tradeoff: ~4/3 = [497.63067, 498.04500]

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.53

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262

Mapping: [⟨1 0 80 25 -33 -93 -137], ⟨0 1 -49 -14 23 61 89]]

Optimal tunings:

  • WE: ~2 = 1199.8939 ¢, ~3/2 = 702.2445 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3064 ¢

Tuning ranges:

  • 17-odd-limit diamond monotone: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
  • 17-odd-limit diamond tradeoff: ~4/3 = [497.63067, 498.04500]

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.50

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728

Mapping: [⟨1 0 80 25 -33 -93 -137 74], ⟨0 1 -49 -14 23 61 89 -44]]

Optimal tunings:

  • WE: ~2 = 1199.8766 ¢, ~3/2 = 702.2355 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3077 ¢

Tuning ranges:

  • 19- and 21-odd-limit diamond monotone: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
  • 19- and 21-odd-limit diamond tradeoff: ~4/3 = [497.62290, 498.04500]

Optimal ET sequence: 41, 176, 217

Badness (Sintel): 1.33

Newt

For the 5-limit version, see Schismic–countercommatic equivalence continuum #Newt (5-limit).

Newt tempers out the breedsma and may be described as the 41 & 270 temperament. It has a generator of a neutral third (0.2 cents flat of 49/40) with a ploidacot signature of dicot. 41 generator steps fall short of 12 octaves by a generic aberschisma step of a schisma~aberschisma. From there the intervals of 5 and 7 can be derived.

Like cotoneum, newt can be notated in the same way as cassaschismic, but with half-sharps and half-flats and the extra equivalence that two comma steps and an aberschisma step make a half-apotome step. In other words, C–^↑↑E = C–v↓↓E = C–Ed.

Newt continues to be significant as an 11-limit temperament, where it tempers out the lehmerisma (3025/3024). This extends into a very strong 13-limit temperament and eventually a very strong no-17 19-limit temperament, a.k.a. neonewt. 270edo and 311edo are obvious tuning choices, but 581edo and especially 851edo are more accurate.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]

mapping generators: ~2, ~49/40

Optimal tunings:

  • WE: ~2 = 1199.9315 ¢, ~49/40 = 351.0932 ¢
error map: ⟨-0.068 +0.163 +0.075 -0.188]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1141 ¢
error map: ⟨0.000 +0.273 +0.180 -0.022]

Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201

Badness (Sintel): 1.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]

Optimal tunings:

  • WE: ~2 = 1199.9603 ¢, ~49/40 = 351.1038 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1155 ¢

Optimal ET sequence: 41, 188, 229, 270, 581, 851, 1121, 1972

Badness (Sintel): 0.643

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 19 11 -10 -20], ⟨0 2 -57 -28 46 81]]

Optimal tunings:

  • WE: ~2 = 1199.9747 ¢, ~49/40 = 351.1094 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1168 ¢

Optimal ET sequence: 41, 229, 270, 581, 851, 2283b

Badness (Sintel): 0.571

2.3.5.7.11.13.19 subgroup (neonewt)

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400

Mapping: [⟨1 1 19 11 -10 -20 18], ⟨0 2 -57 -28 46 81 -47]]

Optimal tunings:

  • WE: ~2 = 1199.9782 ¢, ~49/40 = 351.1102 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1166 ¢

Optimal ET sequence: 41, 229, 270, 581, 851

Badness (Sintel): 0.438

Gariwizmic

Gariwizmic tempers out the wizma and the garischisma, and may be described as the 94 & 176 temperament. It assumes a semioctave period and a perfect fifth generator that is slightly sharp of just. It finds 5/4 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup. Notable tunings not appearing in the optimal ET sequence include 364edo and 634edo.

Gariwizmic was named by Eufalesio in 2026 as a concatenation of gary and wizmic.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 33554432/33480783

Mapping: [⟨2 0 -119 50], ⟨0 1 39 -14]]

mapping generators: ~46305/32768, ~3

Optimal tunings:

  • WE: ~46305/32768 = 599.9657 ¢, ~3/2 = 702.1765 ¢
error map: ⟨-0.069 +0.153 -0.021 -0.053]
  • CWE: ~46305/32768 = 600.0000 ¢, ~3/2 = 702.2161 ¢
error map: ⟨0.000 +0.261 +0.114 +0.149]

Optimal ET sequence: 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd

Badness (Sintel): 2.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 19712/19683, 41503/41472

Mapping: [⟨2 0 -119 50 -66], ⟨0 1 39 -14 23]]

Optimal tunings:

  • WE: ~99/70 = 599.9790 ¢, ~3/2 = 702.1938 ¢
error map: ⟨-0.042 +0.197 +0.106 -0.001 -0.440]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2179 ¢
error map: ⟨0.000 +0.263 +0.185 +0.123 -0.306]

Optimal ET sequence: 94, 176, 270, 1174, 1444, 1714, 1984e

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683

Mapping: [⟨2 0 -119 50 -66 93], ⟨0 1 39 -14 23 -27]]

Optimal tunings:

  • WE: ~99/70 = 599.9958 ¢, ~3/2 = 702.2096 ¢
error map: ⟨-0.008 +0.246 +0.035 +0.146 -0.412 -0.353]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2145 ¢
error map: ⟨0.000 +0.260 +0.054 +0.170 -0.383 -0.321]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.822

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079

Mapping: [⟨2 0 -119 50 -66 93 -131], ⟨0 1 39 -14 23 -27 44]]

Optimal tunings:

  • WE: ~99/70 = 599.9969 ¢, ~3/2 = 702.2114 ¢
error map: ⟨-0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2150 ¢
error map: ⟨0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.655

Satin

For the 5-limit version, see Miscellaneous 5-limit temperaments #Satin.

Satin tempers out the rainy comma and the canousma in addition to the garischisma, and may be described as the 94 & 217 temperament. It uses ~11/10 as a generator, three of which gives a perfect fourth, tempering out 4000/3993 in the 11-limit and onwards. Its ploidacot is omega-tricot.

Subgroup: 2.3.5.7

Comma list: 2100875/2097152, 4802000/4782969

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

mapping generators: ~2, ~8575/7776

Optimal tunings:

  • WE: ~2 = 1200.0198 ¢, ~8575/7776 = 165.9161 ¢
error map: ⟨+0.020 +0.336 -0.200 -0.411]
  • CWE: ~2 = 1200.0000 ¢, ~8575/7776 = 165.9133 ¢
error map: ⟨0.000 +0.305 -0.241 -0.469]

Optimal ET sequence: 94, 217, 311, 839, 1150

Badness (Sintel): 4.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 19712/19683, 41503/41472

Mapping: [⟨1 2 12 -3 13], ⟨0 -3 -70 42 -69]]

Optimal tunings:

  • WE: ~2 = 1199.9931 ¢, ~11/10 = 165.9145 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9155 ¢

Optimal ET sequence: 94, 217, 311

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689

Mapping: [⟨1 2 12 -3 13 -1], ⟨0 -3 -70 42 -69 34]]

Optimal tunings:

  • WE: ~2 = 1199.9607 ¢, ~11/10 = 165.9085 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9141 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095

Mapping: [⟨1 2 12 -3 13 -1 11], ⟨0 -3 -70 42 -69 34 -50]]

Optimal tunings:

  • WE: ~2 = 1199.9843 ¢, ~11/10 = 165.9110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9132 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 1.02

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573

Mapping: [⟨1 2 12 -3 13 -1 11 16], ⟨0 -3 -70 42 -69 34 -50 -85]]

Optimal tunings:

  • WE: ~2 = 1199.9875 ¢, ~11/10 = 165.9111 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9129 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 0.881

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155

Mapping: [⟨1 2 12 -3 13 -1 11 16 16], ⟨0 -3 -70 42 -69 34 -50 -85 -83]]

Optimal tunings:

  • WE: ~2 = 1199.9745 ¢, ~11/10 = 165.9103 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9140 ¢

Optimal ET sequence: 94, 217, 311

Badness (Sintel): 0.871

Sextile

For the 5-limit version, see Schismic–commatic equivalence continuum #Sextile (5-limit).

Sextile tempers out the landscape comma with a 1/6-octave period and is the 12 & 270 temperament.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 33554432/33480783

Mapping: [⟨6 0 71 150], ⟨0 1 -6 -14]]

mapping generators: ~4096/3645, ~3

Optimal tunings:

  • WE: ~4096/3645 = 199.9828 ¢, ~3/2 = 702.1521 ¢
error map: ⟨-0.103 +0.094 +0.173 -0.088]
  • CWE: ~4096/3645 = 200.0000 ¢, ~3/2 = 702.2187 ¢
error map: ⟨0.000 +0.264 +0.374 +0.112]

Optimal ET sequence: 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc

Badness (Sintel): 1.77

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 151263/151250

Mapping: [⟨6 0 71 150 230], ⟨0 1 -6 -14 -22]]

Optimal tunings:

  • WE: ~55/49 = 199.9817 ¢, ~3/2 = 702.1383 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2080 ¢

Optimal ET sequence: 12, …, 258e, 270, 822, 1092, 1362c

Badness (Sintel): 0.981

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647

Mapping: [⟨6 0 71 150 230 279], ⟨0 1 -6 -14 -22 -27]]

Optimal tunings:

  • WE: ~55/49 = 199.9804 ¢, ~3/2 = 702.1260 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2001 ¢

Optimal ET sequence: 12f, …, 258ef, 270, 552, 822, 1092, 1914cde

Badness (Sintel): 0.788

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625

Mapping: [⟨6 0 71 150 230 279 35], ⟨0 1 -6 -14 -22 -27 -1]]

Optimal tunings:

  • WE: ~55/49 = 199.9826 ¢, ~3/2 = 702.1359 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2003 ¢

Optimal ET sequence: 12f, 258ef, 270, 552, 822, 1092

Badness (Sintel): 0.634

World calendar

World calendar tempers out the dimcomp comma and the garischisma, and can be described as the 12 & 364 temperament. The name derives from a certain calendar layout by the same name.

Subgroup: 2.3.5.7

Comma list: 390625/388962, 33554432/33480783

Mapping: [⟨4 1 -44 86], ⟨0 2 -13 -28]]

mapping generators: ~25/21, ~91125/57344

Optimal tunings:

  • WE: ~25/21 = 299.9938 ¢, ~91125/57344 = 801.0780 ¢
error map: ⟨-0.025 +0.195 -0.603 +0.452]
  • CWE: ~25/21 = 300.0000 ¢, ~91125/57344 = 801.0955 ¢
error map: ⟨0.000 +0.236 -0.555 +0.501]

Optimal ET sequence: 12, …, 352, 364

Badness (Sintel): 7.39

2.3.5.7.17 subgroup

Subgroup: 2.3.5.7.17

Comma list: 2025/2023, 24576/24565, 390625/388962

Subgroup-val mapping: [⟨4 1 -44 86 3], ⟨0 2 -13 -28 5]]

Optimal tunings:

  • WE: ~25/21 = 299.9861 ¢, ~27/17 = 801.0536 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/17 = 801.0919 ¢

Optimal ET sequence: 12, …, 352, 364

Badness (Sintel): 2.74

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000

Subgroup-val mapping: [⟨4 1 -44 86 3 25], ⟨0 2 -13 -28 5 -3]]

Optimal tunings:

  • WE: ~25/21 = 299.9982 ¢, ~27/17 = 801.0898 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/17 = 801.0946 ¢

Optimal ET sequence: 12, …, 352, 364

Badness (Sintel): 1.82

Heptacot

For the 5-limit version, see Schismic–commatic equivalence continuum #Heptacot (5-limit).

Heptacot tempers out the meter and may be described as the 12 & 311 temperament, splitting the perfect fifth into seven semitones. It is the natural 7-limit extension of the 5-limit temperament named by Tristan Bay in 2024. 311edo and 323edo are obvious tuning choices, as well as anything in between such as 634edo.

Heptacot extends to the 11-limit in the same way as does gary, which best preserves its accuracy, though it should be noted that 299 & 311 and 323 & 335d make for simpler but less accurate alternative extensions.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 33554432/33480783

Mapping: [⟨1 1 6 11], ⟨0 7 -44 -98]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1199.9434 ¢, ~1323/1250 = 100.3096 ¢
error map: ⟨-0.057 +0.155 -0.274 +0.215]
  • CWE: ~2 = 1200.0000 ¢, ~1323/1250 = 100.3148 ¢
error map: ⟨0.000 +0.249 -0.165 +0.324]

Optimal ET sequence: 12, …, 299, 311, 323, 634, 957, 1591

Badness (Sintel): 3.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 41503/41472, 703125/702464

Mapping: [⟨1 1 6 11 -10], ⟨0 7 -44 -98 161]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1199.9981 ¢, ~1323/1250 = 100.3174 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1323/1250 = 100.3176 ¢

Optimal ET sequence: 12e, 311, 634, 945

Badness (Sintel): 3.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 19712/19683, 31250/31213

Mapping: [⟨1 1 6 11 -10 -7], ⟨0 7 -44 -98 161 128]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1199.9938 ¢, ~675/637 = 100.3169 ¢
  • CWE: ~2 = 1200.0000 ¢, ~675/637 = 100.3174 ¢

Optimal ET sequence: 12e, 311, 634, 945

Badness (Sintel): 1.89

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 31250/31213

Mapping: [⟨1 1 6 11 -10 -7 5], ⟨0 7 -44 -98 161 128 -9]]

mapping generators: ~2, ~1323/1250

Optimal tunings:

  • WE: ~2 = 1200.0076 ¢, ~675/637 = 100.3179 ¢
  • CWE: ~2 = 1200.0000 ¢, ~675/637 = 100.3173 ¢

Optimal ET sequence: 12e, 311, 634, 945

Badness (Sintel): 1.38

Garitritonic

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Garitritonic may be described as the 53 & 581 temperament, splitting the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. 634edo makes for a strong 7-limit tuning, though in the higher limits one may prefer sticking to 581edo.

Garitritonic was named by Flora Canou in 2026 as a contraction of gary and tritonic.

Subgroup: 2.3.5.7

Comma list: 33554432/33480783, 95703125/95551488

Mapping: [⟨1 -3 -15 67], ⟨0 9 34 -126]]

mapping generators: ~2, ~4375/3072

Optimal tunings:

  • WE: ~2 = 1199.9678 ¢, ~4375/3072 = 611.3417 ¢
error map: ⟨-0.032 +0.217 -0.213 -0.036]
  • CWE: ~2 = 1200.0000 ¢, ~4375/3072 = 611.3582 ¢
error map: ⟨0.000 +0.268 -0.136 +0.045]

Optimal ET sequence: 53, 422d, 475, 528, 581, 634, 1215

Badness (Sintel): 6.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 41503/41472, 1953125/1948617

Mapping: [⟨1 -3 -15 67 -102], ⟨0 9 34 -126 207]]

Optimal tunings:

  • WE: ~2 = 1199.9795 ¢, ~4375/3072 = 611.3485 ¢
  • CWE: ~2 = 1200.0000 ¢, ~4375/3072 = 611.3589 ¢

Optimal ET sequence: 53, 528, 581, 1796, 2377b

Badness (Sintel): 3.60

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 19712/19683, 78125/78078

Mapping: [⟨1 -3 -15 67 -102 -34], ⟨0 9 34 -126 207 74]]

Optimal tunings:

  • WE: ~2 = 1199.9813 ¢, ~500/351 = 611.3494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~500/351 = 611.3589 ¢

Optimal ET sequence: 53, 528, 581, 1796, 2377b

Badness (Sintel): 1.73

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 59375/59319

Mapping: [⟨1 -3 -15 67 -102 -34 -36], ⟨0 9 34 -126 207 74 79]]

Optimal tunings:

  • WE: ~2 = 1199.9884 ¢, ~500/351 = 611.3531 ¢
  • CWE: ~2 = 1200.0000 ¢, ~500/351 = 611.3590 ¢

Optimal ET sequence: 53, 528, 581, 1796, 2377b

Badness (Sintel): 1.22

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